Timing recovery for optical coherent receivers in the presence of polarization mode dispersion

ABSTRACT

A timing recovery system generates a sampling clock to synchronize sampling of a receiver to a symbol rate of an incoming signal. The input signal is received over an optical communication channel. The receiver generates a timing matrix representing coefficients of a timing tone detected in the input signal. The timing tone representing frequency and phase of a symbol clock of the input signal and has a non-zero timing tone energy. The receiver computes a rotation control signal based on the timing matrix that represents an amount of accumulated phase shift in the input signal relative to the sampling clock. A numerically controlled oscillator is controlled to adjust at least one of the phase and frequency of the sampling clock based on the rotation control signal.

RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.15/623,292 entitled “Timing Recovery for Optical Coherent Receivers inthe Presence of Polarization Mode Dispersion” filed on Jun. 14, 2017,which is a continuation of U.S. patent application Ser. No. 14/869,676entitled “Timing Recovery for Optical Coherent Receivers in the Presenceof Polarization Mode Dispersion” filed on Sep. 29, 2015, now U.S. Pat.No. 9,712,253, issued on Jul. 18, 2017, which is a continuation of U.S.patent application Ser. No. 14/095,789 entitled “Timing Recovery forOptical Coherent Receivers in the Presence of Polarization ModeDispersion” filed on Dec. 3, 2013, now U.S. Pat. No. 9,178,625, issuedon Nov. 3, 2015, which claims the benefit of the follow U.S. ProvisionalApplications: U.S. Provisional Application No. 61/732,885 entitled“Timing Recovery for Optical Coherent Receivers In the Presence ofPolarization Mode Dispersion” filed on Dec. 3, 2012 to Mario R. Hueda,et al.; U.S. Provisional Application No. 61/749,149 entitled “TimingRecovery for Optical Coherent Receivers In the Presence of PolarizationMode Dispersion” filed on Jan. 4, 2013 to Mario R. Hueda, et al.; U.S.Provisional Application No. 61/832,513 entitled “Timing Recovery forOptical Coherent Receivers In the Presence of Polarization ModeDispersion” filed on Jun. 7, 2013 to Mario R. Hueda, et al.; and U.S.Provisional Application No. 61/893,128 entitled “Timing Recovery forOptical Coherent Receivers In the Presence of Polarization ModeDispersion” filed on Oct. 18, 2013 to Mario R. Hueda, et al. Thecontents of each of the above referenced applications are incorporatedby reference herein.

BACKGROUND 1. Field of the Art

The disclosure relates generally to communication systems, and morespecifically, to timing recovery in an optical receiver.

2. Description of the Related Art

The most recent generation of high-speed optical transport networksystems has widely adopted receiver technologies with electronicdispersion compensation (EDC). In coherent as well as in intensitymodulation direct detection (IM-DD) receivers, EDC mitigates fiberimpairments such as chromatic dispersion (CD) and polarization modedispersion (PMD). Timing recovery (TR) in the presence of differentialgroup delay (DGD) caused by PMD has been identified as one of the mostcritical challenges for intradyne coherent receivers. This can result inthe receiver failing to recover data received over the fiber channel,thereby decreasing performance of the optical network system.

SUMMARY

A receiver performs a timing recovery method to recover timing of aninput signal. The input signal is received over an optical communicationchannel. In an embodiment, the optical channel may introduce animpairment into the input signal including at least one: a half symbolperiod differential group delay, a cascaded differential group delay, adynamic polarization mode dispersion, and a residual chromaticdispersion. The receiver samples the input signal based on a samplingclock. The receiver generates a timing matrix representing coefficientsof a timing tone detected in the input signal. The timing tonerepresenting frequency and phase of a symbol clock of the input signaland has a non-zero timing tone energy. The receiver computes a rotationcontrol signal based on the timing matrix that represents an accumulatedphase shift of the input signal relative to the sampling clock. Anumerically controlled oscillator is controlled to generate the samplingclock based on the rotation control signal.

In one embodiment, the receiver includes a resonator filter to filterthe input signal to generate a band pass filtered signal. An in-phaseand quadrature error signal is computed based on the band pass filtersignal. The in-phase and quadrature error signal represents an amount ofphase error in each of an in-phase component and a quadrature componentof the input signal. The rotation control signal is computed based onthe in-phase and quadrature error signal. In various embodiments, adeterminant method or a modified wave difference method can be used todetermine the rotation control signal based on the timing matrix.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention has other advantages and features which will be morereadily apparent from the following detailed description of theinvention and the appended claims, when taken in conjunction with theaccompanying drawings, in which:

FIG. 1 is a system diagram of an embodiment of an optical communicationsystem.

FIG. 2 is a plot illustrating an effect of half baud DGD in anuncompensated optical receiver where a single polarization is used fortiming recovery.

FIG. 3 is a plot illustrating an effect of half baud DGD in anuncompensated optical receiver where two polarizations are used fortiming recovery.

FIG. 4 is a diagram modeling PMD effects in an optical channel.

FIG. 5 is a block diagram illustrating an embodiment of a timingrecovery system.

FIG. 6 is a block diagram illustrating an embodiment of a resonatorfilter for a timing recovery system.

FIG. 7 is a block diagram illustrating an embodiment of an in-phase andquadrature error computation block filter for a timing recovery system.

FIG. 8A is a block diagram illustrating an embodiment of a phase errorcomputation block for a timing recovery system that applies adeterminant method.

FIG. 8B is a block diagram illustrating an embodiment of a phase errorcomputation block for a timing recovery system that applies a modifiedwave difference method.

DETAILED DESCRIPTION Overview

A receiver architecture and method for timing recovery is described fortransmissions received over an optical fiber channel in the presence ofdifferential group delay (DGD) (caused, for example, by polarizationmode dispersion) that affects the detectability of a reliable timingtone. Timing recovery can then be performed on the transformed signal torecover a clock signal.

High Level System Architecture

FIG. 1 is a block diagram of a communication system 100. Thecommunication system 100 comprises a transmitter 110 for encoding dataas an electrical signal, an optical transmitter 120 for converting theelectrical signal produced by the transmitter 110 to an optical signalsuitable for transmission over a communication channel 130, an opticalfront end 150 for converting the received optical signal to anelectrical signal, and a receiver for receiving and processing theelectrical signal encoding the data from the optical front end 150. Inone embodiment, the communication system 100 comprises an ultra-highspeed (e.g., 40 Gb/s or faster) optical fiber communication system,although the described techniques may also be applicable to lower speedoptical communication systems.

The transmitter 110 comprises an encoder 112, a modulator 114, atransmitter (Tx) digital signal processor (DSP) 116, and Tx analog frontend (AFE) 118. The encoder 112 receives input data 105 and encodes thedata for transmission over the optical network. For example, in oneembodiment, the encoder 112 encodes the input data 105 using forwarderror correction (FEC) codes that will enable the receiver 160 todetect, and in many cases, correct errors in the data received over thechannel 130. The modulator 114 modulates the encoded data via one ormore carrier signals for transmission over the channel 130. For example,in one embodiment, the modulator 114 applies phase-shift keying (PSK) ordifferential phase-shift keying (DPSK) to the encoded data. The Tx DSP116 adapts (by filtering, etc.) the modulator's output signal accordingto the channel characteristics in order to improve the overallperformance of the transmitter 110. The Tx AFE 118 further processes andconverts the Tx DSP's digital output signal to the analog domain beforeit is passed to the optical transmitter (Optical Tx) 120 where it isconverted to an optical signal and transmitted via the channel 130. Oneexample of the optical transmitter 120 transmits independent modulationson both polarizations of the optical carrier. An example modulation isQPSK, though other modulations can be used, and the choice can be madeto transmit on either one or both polarizations.

In addition to the illustrated components, the transmitter 110 maycomprise other conventional features of a transmitter 110 which areomitted from FIG. 1 for clarity of description. Furthermore, in oneembodiment, the transmitter 110 is embodied as a portion of atransceiver device that can both transmit and receive over the channel130.

The channel 130 may have a limited frequency bandwidth and may act as afilter on the transmitted data. Transmission over the channel 130 mayadd noise to the transmitted signal including various types of randomdisturbances arising from outside or within the communication system100. Furthermore, the channel 130 may introduce fading and/orattenuation effects to the transmitted data. Additionally, the channel130 may introduce chromatic dispersion (CD) and polarization modedispersion (PMD) effects that cause a spreading of pulses in the channel130. Based on these imperfections in the channel 130, the receiver 160is designed to process the received data and recover the input data 105.

In general, the optical front end 150 receives the optical signal,converts the optical signal to an electrical signal, and passes theelectrical signal to the receiver 160. The receiver 160 receives theencoded and modulated data from the transmitter 110 via the opticaltransmitter 120, communication channel 130, and optical front end 150,and produces recovered data 175 representative of the input data 105.The receiver 160 includes a receiver (Rx) analog front end (AFE) 168, anRX DSP 166, a demodulator 164, and a decoder 162. The Rx AFE 168 samplesthe analog signal from the optical front end 150 based on a clock signal181 to convert the signal to the digital domain. The Rx DSP 166 furtherprocesses the digital signal by applying one or more filters to improvesignal quality. As will be discussed in further detail below, the Rx DSP166 includes a timing recovery block 179 that operates to generate thesampling clock 181 and to adjust the sampling frequency and phase of thesampling clock signal 181 to ensure that the sampling clock remainssynchronized with the symbol rate and phase of the incoming opticalsignal. This timing recovery problem becomes challenging due to theimperfections in the channel 130 that may alter the received opticalsignal. For example, chromatic dispersion (CD) and polarization modedispersion (PMD) effects may cause a spreading of pulses in the channel130, thereby increasing the difficulty of timing recovery, as will beexplained below.

The demodulator 164 receives the modulated signal from the Rx DSP 166and demodulates the signal. The decoder 162 decodes the demodulatedsignal (e.g., using error correction codes) to recover the originalinput data 105.

In addition to the illustrated components, the receiver 160 may compriseother conventional features of a receiver 160 which are omitted fromFIG. 1 for clarity of description. Furthermore, in one embodiment, thereceiver 160 is embodied as a portion of a transceiver device that canboth transmit and receive over the channel 130.

Components of the transmitter 110 and the receiver 160 described hereinmay be implemented, for example, as an integrated circuit (e.g., anApplication-Specific Integrated Circuit (ASIC) or using afield-programmable gate array (FPGA), in software (e.g., loading programinstructions to a processor (e.g., a digital signal processor (DSP))from a computer-readable storage medium and executing the instructionsby the processor), or by a combination of hardware and software.

Impact of Channel Impairments on Timing Recovery

In order for the timing recovery block 179 to properly generate thesampling clock 181, a timing tone is detected in the digital inputsignal that will ideally appear at a frequency representative of thesymbol rate. A common technique for timing recovery is the nonlinearspectral line method as described in J. R. Barry, E. A. Lee, and D. G.Messerschmitt, Digital Communication. KAP, third ed. 2004. In thismethod, a timing tone is detected in the digital input signal and thefrequency of the timing tone is used to synchronize the sampling clock181 to the incoming signal. However, under certain conditions, thetiming tone can be lost or offset when using a conventional spectralline method timing recovery technique. For example, the timing tone maydisappear in the presence of half-baud DGD. Furthermore, when a cascadedDGD is present, the nonlinear spectral line method may generate a timingtone with a frequency offset which causes a loss of synchronization.Other channel conditions such as general dynamic PMD and/or residualchromatic dispersion can furthermore cause the traditional spectral linemethod to fail due to frequency or phase offset between the symbol clockand the sampling clock.

The following description explains how the PMD is expressedmathematically, which provides a basis for an explanation of the effectsof particular PMD on the timing information in the received signal. Thesignal from the optical front end 150 presented to the receiver 160consists of electrical signals from both polarizations of the opticalsignal received by the optical front end 150. These two polarizationscan be treated mathematically as a two-dimensional complex vector, whereeach component corresponds to one of the polarizations of the receivedoptical signal. Alternatively, the two polarizations can be treatedmathematically as a four-dimensional real vector, where two of thedimensions correspond to the in-phase and quadrature components of onepolarization, and the other two components correspond to the in-phaseand quadrature components of the other polarization.

This example demonstrates the embodiment where the two polarizations atthe transmitter are each modulated independently. Let {a_(k)} and{b_(k)} respectively represent the symbol sequences transmitted on thehorizontal and vertical polarizations of the optical signal, where thesymbols are in general complex. For this example, it is assumed thata_(k) and b_(k) are independent and identically distributed complex datasymbols with E{a_(k)a*_(m)}=E{b_(k)b*_(m)}=δ_(m-k) where δ_(k) is thediscrete time impulse function and E{.} denotes expected value. Let X(ω)be the Fourier transform of the channel input

${\sum\limits_{k}\; {\begin{bmatrix}a_{k} \\b_{k}\end{bmatrix}{\delta \left( {t - {kT}} \right)}}},$

where δ(t) is the continuous time impulse function (or delta function),and T is the symbol period (also called one baud). Let S(ω) be theFourier transform of the transmit pulse, s(t). In the presence of CD andPMD, the channel output can be written as H(ω)X(ω), with the channeltransfer matrix expressed as

H(ω)=e ^(−jβ(ω)L) J(ω)S(ω),  (1)

where ω is the angular frequency, L is the fiber length of the opticalchannel 130, β(ω) is the CD parameter,

$\begin{matrix}{{{S(\omega)} = \begin{bmatrix}{S(\omega)} & 0 \\0 & {S(\omega)}\end{bmatrix}},} & (2)\end{matrix}$

and J(ω) is the Jones matrix. The components of J(ω) are defined by

$\begin{matrix}{{{J(\omega)} = \begin{bmatrix}{U(\omega)} & {V(\omega)} \\{- {V^{*}(\omega)}} & {U^{*}(\omega)}\end{bmatrix}},} & (3)\end{matrix}$

where * denotes complex conjugate. Matrix J(ω) is special unitary (i.e.,J(ω)^(H)J(ω)=I, det(J(ω))=1, where I is the 2×2 identity matrix and Hdenotes conjugate transpose) and models the effects of the PMD. Forexample, the Jones matrix for first-order PMD reduces to

$\begin{matrix}{{J(\omega)} = {{R\left( {\theta_{0}\varphi_{0}} \right)}\begin{bmatrix}e^{{j\; \omega \frac{\tau}{2}} + {j\frac{\psi_{0}}{2}}} & 0 \\0 & e^{{{- j}\; \omega \frac{\tau}{2}} - {j\frac{\psi_{0}}{2}}}\end{bmatrix}}} & (4)\end{matrix}$

where τ is the differential group delay (DGD), ψ₀ is the polarizationphase, and R(.,.) is the rotation matrix given by

$\begin{matrix}{{{R\left( {\theta_{0}\varphi_{0}} \right)}\begin{bmatrix}e^{j\; \frac{\varphi}{2}} & 0 \\0 & e^{{- j}\; \frac{\varphi}{2}}\end{bmatrix}}\begin{bmatrix}{\cos (\theta)} & {\sin (\theta)} \\{- {\sin (\theta)}} & {\cos (\theta)}\end{bmatrix}} & \left( {5a} \right)\end{matrix}$

where θ is the polarization angle and φ is a random phase angle. Notethat first order DGD may be more generally expressed as:

$\begin{matrix}{{{R\left( {\theta_{0}\varphi_{0}} \right)}\begin{bmatrix}e^{{j\; \omega \frac{\tau}{2}} + {j\frac{\psi_{0}}{2}}} & 0 \\0 & e^{{{- j}\; \omega \frac{\tau}{2}} - {j\frac{\psi_{0}}{2}}}\end{bmatrix}}{R^{- 1}\left( {\theta_{1}\varphi_{1}} \right)}} & \left( {5b} \right)\end{matrix}$

but the last rotation matrix does not affect the strength of the timingtone, so the simpler form given in (4) is used herein.

Assuming that CD is completely compensated in the receiver Rx DSP 166 ofthe receiver 160 or prior to the signal reaching the receiver Rx DSP166, (i.e., H(ω)=J(ω)S(ω)), the noiseless signal entering the timingrecovery block 179 can be expressed as:

$\begin{matrix}{{{r(t)} = {\begin{bmatrix}{r_{x}(t)} \\{r_{y}(t)}\end{bmatrix} = \begin{bmatrix}{{\sum_{k = {- \infty}}^{\infty}{{h_{11}\left( {t - {kT}} \right)}a_{k}}} + {{h_{12}\left( {t - {kT}} \right)}b_{k}}} \\{{\sum_{k = {- \infty}}^{\infty}{{h_{21}\left( {t - {kT}} \right)}a_{k}}} + {{h_{22}\left( {t - {kT}} \right)}b_{k}}}\end{bmatrix}}},} & (6)\end{matrix}$

where 1/T is the symbol rate, while

h ₁₁(t)=

⁻¹{U(ω)S(ω)},

h ₁₂(t)=

⁻¹{V(ω)S(ω)},

h ₂₁(t)=

⁻¹{−V*(ω)S(ω)},

h ₂₂(t)=

⁻¹{U*(ω)S(ω)},  (7)

where

⁻¹{.} denotes the inverse Fourier transform.

In the nonlinear spectral line method for timing recovery, the timingrecovery block 179 processes the received signal by a memorylessnonlinearity in order to generate a timing tone with frequency 1/T. Amagnitude squared nonlinearity is used as the memoryless nonlinearityapplied to the received signal. Then, the mean value of the magnitudesquared of the received signal is periodic with period T and can beexpressed through a Fourier series as

$\begin{matrix}{{{E\left\{ {{r_{x}(t)}}^{2} \right\}} = {\frac{1}{\pi \; T}{\sum_{k = 0}^{\infty}{{Re}\left\{ {z_{x,k}e^{j\frac{2\pi \; {kt}}{T}}} \right\}}}}},} & (8) \\{{{E\left\{ {{r_{y}(t)}}^{2} \right\}} = {\frac{1}{\pi \; T}{\sum_{k = 0}^{\infty}{{Re}\left\{ {z_{y,k}e^{j\frac{2\pi \; {kt}}{T}}} \right\}}}}},} & (9)\end{matrix}$

where

$\begin{matrix}{z_{x,k} = {{\int_{- \infty}^{\infty}{{U(\omega)}{U^{*}\left( {\omega - \frac{2\pi \; k}{T}} \right)}{S(\omega)}{S^{*}\left( {\omega - \frac{2\pi \; k}{T}} \right)}d\; \omega}} + {\int_{- \infty}^{\infty}{{V(\omega)}{V^{*}\left( {\omega - \frac{2\pi \; k}{T}} \right)}{S(\omega)}{S^{*}\left( {\omega - \frac{2\pi \; k}{T}} \right)}d\; \omega}}}} & (10) \\{z_{y,k} = {{\int_{- \infty}^{\infty}{{V^{*}(\omega)}{V\left( {\omega - \frac{2\pi \; k}{T}} \right)}{S(\omega)}{S^{*}\left( {\omega - \frac{2\pi \; k}{T}} \right)}d\; \omega}} + {\int_{- \infty}^{\infty}{{U^{*}(\omega)}{U\left( {\omega - \frac{2\pi \; k}{T}} \right)}{S(\omega)}{S^{*}\left( {\omega - \frac{2\pi \; k}{T}} \right)}d\; {\omega.}}}}} & (11)\end{matrix}$

The timing information can be extracted either from the periodic signalderived from the received x polarization (E{|r_(x)(t)|²}), or from theperiodic signal derived from the received y polarization(E{|r_(y)(t)|²}), or from some combination of these two periodicsignals. Additional details regarding the nonlinear spectral line methodfor timing recovery is described in J. R. Barry, E. A. Lee, and D. G.Messerschmitt, Digital Communication. KAP, third ed. 2004.

Timing Information Using One Polarization

The following description explains the effects of PMD when onepolarization is used for timing recovery and gives example conditionswhere the timing recovery information can disappear. From (8)-(9), itcan be seen that the clock signal 181 will be different from zero in|r_(x)(t)|² (or |r_(y)(t)|²) if the magnitude of the Fourier coefficient|z_(x,1)|>0 (or |z_(y,1)|>0). On the other hand, from (10) and (11) itcan be verified that the timing tone in each polarization component willbe zero if

$\begin{matrix}{{{V(\omega)} = {e^{j\; \psi}e^{{\pm j}\; \omega \frac{T}{2}}{U(\omega)}}},} & (12)\end{matrix}$

where ψ is an arbitrary phase. Since the Jones matrix is special unitary(so |U(ω)|²+|V(ω)|²=1), the following expression can be derived from(12)

$\begin{matrix}{{{U(\omega)}} = {{{V(\omega)}} = {\frac{1}{\sqrt{2}}.}}} & (13)\end{matrix}$

Let γ(ω)/2 be the phase response of U(ω). Then, from (12) and (13), theclock signal disappears when the Jones matrix representing the PMD atthe input of the timing recovery block 179 can be expressed as

$\begin{matrix}{{{J_{0}(\omega)} = {{\begin{bmatrix}{\frac{1}{\sqrt{2}}e^{j\frac{\gamma {(\omega)}}{2}}} & {\frac{1}{\sqrt{2}}e^{j\frac{\gamma {(\omega)}}{2}}} \\{{- \frac{1}{\sqrt{2}}}e^{{- j}\frac{\gamma {(\omega)}}{2}}} & {\frac{1}{\sqrt{2}}e^{{- j}\frac{\gamma {(\omega)}}{2}}}\end{bmatrix}\begin{bmatrix}e^{{{\pm j}\; \omega \frac{T}{4}} + {j\frac{\psi}{2}}} & 0 \\0 & e^{{{\mp j}\; \omega \frac{T}{4}} - {j\frac{\psi}{2}}}\end{bmatrix}} = {R{{\left( {{\pi/4},{\gamma (\omega)}} \right)\begin{bmatrix}e^{{{\pm j}\; \omega \frac{T}{4}} + {j\frac{\psi}{2}}} & 0 \\0 & e^{{{\mp j}\; \omega \frac{T}{4}} - {j\frac{\psi}{2}}}\end{bmatrix}}.}}}}\quad} & (14)\end{matrix}$

For example, the impact of the first-order PMD defined by (4) isanalyzed in FIG. 2. Here, the plot illustrates the normalized magnitudeof the timing tone coefficient z_(x,1) derived from (10), versus DGD (τ)and the rotation angle (θ) assuming an ideal lowpass pulse s(t) withbandwidth excess <100%. In particular, for τ=T/2, and θ₀=π/4, thefollowing expression can be derived from (4):

$\begin{matrix}{{J(\omega)} = {{{R\left( {{\pi/4},\varphi_{0}} \right)}\begin{bmatrix}e^{{j\; \omega \frac{\tau}{2}} + {j\frac{\psi_{0}}{2}}} & 0 \\0 & e^{{{- j}\; \omega \frac{T}{2}} - {j\frac{\psi_{0}}{2}}}\end{bmatrix}}.}} & (15)\end{matrix}$

The matrix in (15) can be written as (14) (with γ(ω)=φ₀ and ψ=ψ₀). Itcan be inferred from the equations above that the clock signal maydisappear with half-baud DGD, as illustrated in FIG. 2.

Timing Information Using Two Polarizations

The effects of PMD are now described when both polarizations are usedfor timing recovery and conditions where the timing recovery informationcan disappear are presented. In particular, the sum of the squaredsignals of both polarizations may be used for timing recovery (i.e.,|r_(x)(t)|²+|r_(y)(t)|²). Then, the total timing tone coefficientz_(x+y,1) can be expressed as:

$\begin{matrix}{{z_{{x + y},1} = {{z_{x,1} + z_{y,1}} = {{\int_{- \infty}^{\infty}{2\left\{ {{U(\omega)}{U^{*}\left( {\omega - \frac{2\pi}{T}} \right)}} \right\} {S(\omega)}{S^{*}\left( {\omega - \frac{2\pi}{T}} \right)}d\; \omega}} + {\int_{- \infty}^{\infty}{2\left\{ {{V(\omega)}{V^{*}\left( {\omega - \frac{2\pi}{T}} \right)}} \right\} {S(\omega)}{S^{*}\left( {\omega - \frac{2\pi}{T}} \right)}d\; \omega}}}}},} & (16)\end{matrix}$

where

{.} denotes the real part of the expression.

For example, from (16) it is observed that the clock signal will be zeroif

$\begin{matrix}{{{U(\omega)} = {{{{U(\omega)}}e^{{\pm j}\; \omega \frac{T}{4}}e^{j\frac{\psi_{U}}{2}}\mspace{14mu} {and}\mspace{14mu} {V(\omega)}} = {{{V(\omega)}}e^{{\pm j}\; \omega \frac{T}{4}}e^{j\frac{\psi_{V}}{2}}}}},} & (17)\end{matrix}$

where ψ_(U) and ψ_(V) are arbitrary angles. As can be seen, thefirst-order PMD defined by (4) with τ=T/2 satisfies condition (17) forany combination of θ₀, φ₀ and ψ₀. Therefore, the clock signal containedin |r_(x)(t)|²+|r_(y)(t)|² is lost in the presence of half-baud DGD.This is illustrated in FIG. 3, which is a plot of the normalized timingtone magnitude in |r_(x)(t)|²+|r_(y)(t)|² versus DGD (τ) and therotation angle (θ).

Timing Information in Cascaded DGD with Two Segments

Under certain conditions, the timing tone generated using the nonlinearspectral line method may generate a timing tone with a frequency offsetwhich causes a loss of synchronization. An example channel 430 (whichmay be used as channel 130) having these conditions is illustrated inFIG. 4. Here, the channel is modeled as a first DGD block 402, a firstmatrix rotation block 404, a second DGD block 406, and a second matrixrotation block 408, which each represent distortions in the channel. Thesignal sent over the channel 430 comprises an optical signal having twopolarizations (e.g., a horizontal and vertical polarization, or moregenerally, an “X” and “Y” polarization) of the optical signal receivedby the optical front end 150. These two polarizations can be treatedmathematically as a two-dimensional complex vector, where each componentcorresponds to one of the polarizations of the received optical signaland is expressed as a complex representation of in-phase and quadraturesignals. The first DGD block 402 and second DGD block 406 each apply atransform

$\begin{bmatrix}e^{j\; \omega \; {T/4}} & 0 \\0 & e^{{- j}\; \omega \; {T/4}}\end{bmatrix}\quad$

to the incoming signal. The first matrix rotation block 404 and thesecond matrix rotation block 408 each apply a transform

${MR} = \begin{bmatrix}{\cos (\theta)} & {\sin (\theta)} \\{- {\sin (\theta)}} & {\cos (\theta)}\end{bmatrix}$

where in this example, θ=ω_(R)t for the first matrix rotation block 404and, θ=π/4 for the second matrix rotation block 408.

The timing tone coefficients for each polarization (z_(x,1) and z_(y,1))are given by the diagonal elements of the 2×2 matrix

$\begin{matrix}{M_{t} = {\begin{bmatrix}z_{x,1} & z_{xy} \\z_{yx} & z_{y,1}\end{bmatrix} = {\int_{- \infty}^{\infty}{{S(\omega)}S^{H}\left\{ {\omega - \frac{2\; \pi}{T}} \right\} \; {J(\omega)}J^{H}\left\{ {\omega - \frac{2\; \pi}{T}} \right\} d\; \omega}}}} & (18)\end{matrix}$

where S(ω) is the transfer function of the transmit and receive filtersand J(ω) is the Jones matrix. The components of J(ω) are defined by

$\begin{matrix}{{J(\omega)} = \begin{bmatrix}{U(\omega)} & {V(\omega)} \\{- {V^{*}(\omega)}} & {U^{*}(\omega)}\end{bmatrix}} & (19)\end{matrix}$

where * denotes complex conjugate. Matrix J(ω) is special unitary (i.e.,J(ω)^(H)J(ω)=I, det(J(ω))=1, where I is the 2×2 identity matrix and Hdenotes conjugate transpose) and models the effects of the PMD.

Since ω_(R)<<2π/T, the PMD matrix of the channel 430 describedpreviously is given by:

$\begin{matrix}{{J(\omega)} \approx {\frac{1}{\sqrt{2}}\begin{bmatrix}{{{\cos \left( {\omega_{R}t} \right)}e^{j\frac{\omega \; T}{2}}} - {\sin \left( {\omega_{R}t} \right)}} & {{{\cos \left( {\omega_{R}t} \right)}e^{{- j}\frac{\omega \; T}{2}}} + {\sin \left( {\omega_{R}t} \right)}} \\{{{- {\cos \left( {\omega_{R}t} \right)}}e^{j\frac{\omega \; T}{2}}} - {\sin \left( {\omega_{R}t} \right)}} & {{{\cos \left( {\omega_{R}t} \right)}e^{{- j}\frac{\omega \; T}{2}}} - {\sin \left( {\omega_{R}t} \right)}}\end{bmatrix}}} & (20)\end{matrix}$

Assuming that the bandwidth excess is small-moderate (i.e.,S(ω)S^(H)(ω−2π/T) is concentrated around ω=π/T), then the timing matrixM_(t) can be approximated as:

$\begin{matrix}{M_{t} \approx {K_{s}{J\left( \frac{\pi}{T} \right)}{J^{H}\left( {- \frac{\pi}{T}} \right)}}} & (21)\end{matrix}$

where K_(S) is a given complex constant and

$\begin{matrix}{{{J\left( \frac{\pi}{T} \right)} = {\frac{1}{\sqrt{2}}\begin{bmatrix}{j\; e^{j\; \omega_{R}t}} & {{- j}\; e^{j\; \omega_{R}t}} \\{{- j}\; e^{{- j}\; \omega_{R}t}} & {{- j}\; e^{{- j}\; \omega_{R}t}}\end{bmatrix}}}{{J^{H}\left( {- \frac{\pi}{T}} \right)} = {\frac{1}{\sqrt{2}}\begin{bmatrix}{j\; e^{j\; \omega_{R}t}} & {{- j}\; e^{j\; \omega_{R}t}} \\{{- j}\; e^{{- j}\; \omega_{R}t}} & {{- j}\; e^{{- j}\; \omega_{R}t}}\end{bmatrix}}}} & (22)\end{matrix}$

Based on the above equations,

$\begin{matrix}{M_{t} \approx {- {K_{s}\begin{bmatrix}e^{j\; 2\; \omega_{R}t} & 0 \\0 & e^{{- j}\; 2\; \omega_{R}t}\end{bmatrix}}}} & (23)\end{matrix}$

Note that the timing tone energy in a given polarization is maximized.The resulting clock tone in a given polarization results in

$\begin{matrix}\begin{matrix}{{c_{x}(t)} = {2\; \; \left\{ {z_{x,1}e^{j\; 2\; \pi \; {t/T}}} \right\}}} \\{= {{- 2}\; \; \left\{ {K_{s}e^{j\; 2\; \omega_{R}t}e^{j\; 2\; \pi \; {t/T}}} \right\}}} \\{= {{- 2}\; \; \left\{ {K_{s}e^{j\; 2\; {\pi {({{2/T_{R}} + {1/T}})}}t}} \right\}}} \\{{\omega_{R} = {{2\; \pi \; f_{R}} = {2\; {\pi/T_{R}}}}}}\end{matrix} & (24)\end{matrix}$

From the above, it can be observed that the frequency of the clocksignal provided by the spectral line timing recovery algorithm isshifted by 2/T_(R). In other words, the timing tone is modulated by thetime variations of the DGD which is affected by the frequency of therotation matrices between the DGD segments. Therefore, under theseconditions, a proper clock signal with frequency 1/T will not beproperly detected using a conventional spectral line method technique.

The channel 430 having two DGD elements with a rotating element betweenthem is only one example of a channel having dynamic PMD. A more generalchannel with dynamic PMD is modeled as an arbitrarily large number ofDGD elements, with randomly varying rotation angles between the DGDelements. These channels can cause timing recovery methods based on atraditional nonlinear spectral line method to fail by causing afrequency offset (as described above) or by causing randomly varyingphase between the received symbol clock and the sampling clock. Therandomly varying phase can accumulate to cause an offset of multiplesymbol periods between the received signal and the sampling clock,resulting in eventual failure of the receiver once the receiver can nolonger compensate for the shift in timing. This situation is compoundedby the presence of residual chromatic dispersion that reaches the timingrecovery circuit, which can cause the amplitude of the timing tone tofade. The disclosed embodiments solve for these impairments and performrobustly in the presence of general dynamic PMD and in the presence ofresidual chromatic dispersion.

Timing Recovery Architecture and Method

FIG. 5 illustrates an example embodiment of a timing recovery systemthat generates a timing recovery signal without the problems of thespectral line method discussed above. In one embodiment, the timingrecovery block 179 comprises a resonator filter 504, an in-phase andquadrature error computation block 506, a rotation computation block508, and a numerically controlled oscillator (NCO) control block 510.

The resonator filter 504 receives oversampled input samples of the twopolarizations, which may be received directly from an oversamplinganalog-to-digital converter (ADC) or from an interpolator filter betweenthe ADC and the timing recovery block 179. The resonator filter 504filters the input signal by applying, for example, a band pass filterhaving a center frequency of approximately 1/(2T). This center frequencyis used because the timing information in a spectral line timingrecovery scheme is generally within the vicinity of this frequency band.The phase and quadrature error computation block 506 interpolates thefiltered signal (e.g., from two samples per baud to four samples perbaud per polarization) from the resonator filter 504 and determinesin-phase and quadrature error signals from the interpolated signal. Therotation computation block 508 generates a timing matrix representingdetected timing tones and estimates a rotation error based on the timingmatrix. The rotation error signal controls the NCO control block 510which adjusts the sampling phase or frequency, or both, of the samplingclock 181 based on the rotation error. In alternative embodiments, theNCO control block 510 may instead control sampling phase of aninterpolator filter between the ADC and the timing recovery block,rather than controlling sampling phase and frequency of the samplingclock 181 directly.

FIG. 6 illustrates the input and output signals from the resonatorfilter 504 in more detail. The resonator filter 504 receives anoversampled input signal {tilde over (R)} which may be generated by, forexample, an interpolator filter or directly from an analog-to-digitalconverter in the Rx AFE 168. For example, in one embodiment {tilde over(R)} has a sampling rate of 2/T per polarization where T is the periodof the timing tone. In this example, {tilde over (R)} comprises fourvector components:

{tilde over (R)} _(x)(k,0)=[r _(x)(kN _(TR),0),r _(x)(kN _(TR)+1,0), . .. r _(x)(kN _(TR) +N _(TR)−1,0)]

{tilde over (R)} _(x)(k,2)=[r _(x)(kN _(TR),2),r _(x)(kN _(TR)+1,2), . .. r _(x)(kN _(TR) +N _(TR)−1,2)]

{tilde over (R)} _(y)(k,0)=[r _(y)(kN _(TR),0),r _(y)(kN _(TR)+1,0), . .. r _(y)(kN _(TR) +N _(TR)−1,0)]

{tilde over (R)} _(y)(k,2)=[r _(y)(kN _(TR),2),r _(y)(kN _(TR)+1,2), . .. r _(y)(kN _(TR) +N _(TR)−1,2)]  (25)

where {tilde over (R)}_(x)(k, 0) represents an N_(TR) dimensional vectorwith even samples of a first polarization “X”; {tilde over (R)}_(x)(k,2) represents an N_(TR) dimensional vector with odd samples of the firstpolarization “X”; {tilde over (R)}_(y)(k, 0) represents an N_(TR)dimensional vector with odd samples of a second polarization “Y”; and{tilde over (R)}_(y)(k, 2) represents and N_(TR) dimensional vector witheven samples of the second polarization “Y,” and k is an index numberassociated with each set of 4N_(TR) samples. In one embodiment,N_(TR)=64, although different dimensionalities may be used inalternative embodiments. The polarizations “X” and “Y” are usedgenerically to refer to signals derived from two different polarizationsof an optical signal and in one embodiment correspond to horizontal andvertical polarizations respectively, or vice versa.

Each element of the N_(TR) dimensional vectors are in the form r_(a)(n,l) a∈(x, y) where n represents the number of the symbol and l∈[0, 1, 2,3] represents the sampling phase with four sample phases per symbol perpolarization at a sampling rate of 4/T, two of which (0 and 2) are usedat a sampling rate of 2/T. For example, the two samples associated withthe first polarization “X” in the symbol n are represented by r_(x)(n,0) r_(x)(n, 2) when the signal derived from polarization X is sampled ata sampling rate of 2/T. The two samples associated with the secondpolarization “Y” in the symbol n are represented by r_(y)(n, 0) r_(y)(n,2) when the signal derived from polarization Y is sampled at a samplingrate of 2/T.

The resonator filter 504 applies a unit pulse response h_(rf)(n) to theincoming signal {tilde over (R)} where

$\begin{matrix}{{{h_{rf}(n)} = {\frac{2}{N_{rf}}{\sin \left( {n\frac{\pi}{2}} \right)} \times {{rect}_{N_{rf}}(n)}}}{{{rect}_{N_{rf}}(n)} = {\sum\limits_{i = 0}^{N_{rf} - 1}\delta_{n - i}}}} & (28)\end{matrix}$

where N_(rf) represents the number of taps in the resonator filter 504.In this notation, the unit pulse response h_(rf)(n) is based on asampling rate of 2/T. For example, if the input signal is sampled at2/T, then in one embodiment,

$\begin{matrix}{{N_{rf} = {\left. 16\rightarrow\frac{2}{N_{rf}} \right. = 2^{- 3}}}{or}{N_{rf} = {\left. 32\rightarrow\frac{2}{N_{rf}} \right. = 2^{- 4}}}} & (29)\end{matrix}$

The unit pulse response h_(rf)(n) can also be written as:

$\begin{matrix}{{h_{rf}(n)} = {\frac{2}{N_{rf}}\left\lbrack {0,1,0,{- 1},0,1,\ldots}\mspace{14mu} \right\rbrack}} & (30)\end{matrix}$

By filtering the signal derived from polarization a sampled at a rate of2/T, where a∈(x, y), with the unit pulse response h_(rf)(n) above, theresonator filter 504 generates a signal {circumflex over (R)} comprisingfour vector components: {circumflex over (R)}_(x)(k, 0), {circumflexover (R)}_(x)(k, 2), {circumflex over (R)}_(y)(k, 0), {circumflex over(R)}_(y)(k, 2) having the general form:

{circumflex over (R)} _(a)(k,l)=[r _(a)(kN _(TR) ,l), . . . r _(a)(kN_(TR) +N _(TR)−1,l)]a∈(x,y)l∈(0, 2)  (31)

where

$\begin{matrix}{{{{\overset{\_}{r}}_{a}\left( {{{kN}_{TR} + m},l} \right)} = {\frac{2}{N_{rf}}{\sum\limits_{i = 0}^{{N_{rf}/2} - 1}{\left( {- 1} \right)^{i}{r_{a}\left( {{{kN}_{TR} + m - i},l} \right)}}}}}{m \in {\left\lbrack {0,1,\ldots \mspace{14mu},{N_{TR} - 1}} \right\rbrack a} \in {\left( {x,y} \right)\mspace{25mu} l} \in \left( {0,2} \right)}} & (32)\end{matrix}$

An effect of the resonator filter 504 is to band pass filter the signalaround the frequency 1/(2T) where the timing information is contained.Although chromatic dispersion may be compensated for in the Rx DSP 166or by other means prior to the timing recovery block 179, residualchromatic dispersion (that is, chromatic dispersion that has not beencompensated for prior to the timing recovery circuit 179) has the effectof varying the timing tone level particularly in the presence of a highroll-off factor of the transmit pulse. The resonator filter 504 limitsthe bandwidth excess and reduces high fluctuations of the timing tonelevel caused by residual chromatic dispersion, and thereby reduces thedependence of the timing tone level on the chromatic dispersion.

FIG. 7 is block diagram illustrating an example embodiment of thein-phase and quadrature error computation block 506. In the illustratedembodiment, the block 506 comprises an interpolator filter 702, acorrelation computation block 704, and a difference computation block706.

The interpolator filter 702 interpolates the signal {circumflex over(R)} which has two samples per polarization per symbol, to generate aninterpolated signal R having four samples per polarization per symbol.Since the frequency of the timing tone is 1/T, the sampling rate of theinput signal should be higher than 2/T to recover the timing tone. Theinterpolator filter 702 increases the sampling rate (e.g., to 4/T perpolarization) in order to have a sampling rate high enough to accuratelyestimate the timing tone. The signal R generated by the interpolatorfilter 702 has eight components (four for each polarization) havingform:

R _(a)(k,l)=[r _(a)(kN _(TR) ,l), . . . r _(a)(kN _(TR) +N_(TR)−1,l)]a∈(x,y)l∈(0,1,2,3)  (33)

where:

r _(a)(kN _(TR) +m,0)= r _(a)(kN _(TR) +m−d _(if),0)

r _(a)(kN _(TR) +m,1)=Σ_(i=0) ^(N) ^(if) ⁻¹ c ₀(i) r _(a)(kN _(TR)+m−1−i,2) +Σ_(i=0) ^(N) ^(if) ⁻¹ c ₁(i) r _(a)(kN _(TR) +m−i,0)

r _(a)(kN _(TR) +m,2)= r _(a)(kN _(TR) +m−d _(if),2)

r _(a)(kN _(TR) +m,3)=Σ_(i=0) ^(N) ^(if) ⁻¹ c ₀(i) r _(a)(kN _(TR)+m−i,0) +Σ_(i=0) ^(N) ^(if) ⁻¹ c ₁(i) r _(a)(kN _(TR) +m−i,2)  (34)

for a∈(x, y) and m∈[0, 1, . . . , N_(TR)−1] and where 2N_(if) is the tapnumber of the interpolation filter 702 and d_(if) is a delay whered_(if)<N_(if).

Information of a given transmit polarization may be contained in bothreceived polarizations due to fiber channel impairments such as PMD.Thus, in order to recover the timing tones, a correlation of thereceived signals is performed by the correlation computation block 704.The correlation computation block 704 receives the interpolated signal R(e.g., having 8 N_(TR) dimensional vector components) and generatescorrelated signals r having 16 components which are each a scalarcomplex value. For example, in one embodiment, the correlated signalincludes components r_(xx)(k, 0), r_(xy)(k, 0), r_(yx)(k, 0), r_(yy)(k,0), r_(xx)(k, 1), . . . r_(xx)(k, 3), r_(xy)(k, 3), r_(yx)(k, 3),r_(yy)(k, 3) where each component is given by:

r _(ab)(k,i)=R _(a)(k,i)×R _(b) ^(H)(k,i) a,b∈(x,y)  (35)

where (.)^(H) indicates a transpose and conjugate. All of the timinginformation is contained in the correlated signals r at the output ofthe correlation computation block 704.

To extract the amplitude and phases of the four timing tones offrequency 1/T, the difference computation block 706 computes in-phaseerror p and quadrature error q, which are each defined as the differenceof two samples separated by T/2. In one embodiment, the differencecomputation block 706 receives the correlated signal and generates asignal representative of the in-phase and quadrature error in the blockof samples. In one embodiment, the in-phase and quadrature error signalscomprise four in-phase error components: p_(xx)(k), p_(xy)(k),p_(yx)(k), p_(yy)(k) and four quadrature error components: q_(xx)(k),q_(xy)(k), q_(yx)(k), q_(yy)(k).

In one embodiment, the in-phase error components are given by:

p _(ab)(k)=r _(ab)(k,0)−r _(ab)(k,2)a,b∈(x,y)  (36)

and the quadrature error components are given by:

q _(ab)(k)=r _(ab)(k,1)−r _(ab)(k,3)a,b∈(x,y)  (37)

FIGS. 8A and 8B illustrate alternative embodiments of the rotationcomputation block 508, where the alternative embodiments are designated508-A and 508-B in FIGS. 8A and 8B respectively. In the embodiment ofFIG. 8A, the rotation computation block 508-A estimates the phase errorusing a determinant method. In the embodiment of FIG. 8B, the rotationcomputation block 508-B estimates the phase error using a modified wavedifference method. Referring first to FIG. 8A, in one embodiment, therotation computation block 508-A comprises a matrix estimator 802, acycle slip computation block 804, a DTM block 806, and a loop filter808. The timing matrix estimation block 802 estimates a timing matrixfrom the in-phase and quadrature error signals where the timing tonecoefficient for each element of the matrix is the complex number withreal and imaginary parts given by p and q respectively. In oneembodiment, the timing matrix has the following form:

$\begin{matrix}{{M_{t}(k)} = \begin{bmatrix}{M_{t}^{({1,1})}(k)} & {M_{t}^{({1,2})}(k)} \\{M_{t}^{({2,1})}(k)} & {M_{t}^{({2,2})}(k)}\end{bmatrix}} & (38)\end{matrix}$

The timing matrix M_(t)(k) is calculated as follows:

M _(t) ^((1,1))(k)=βM _(t) ^((1,1))(k−1)+(1−β)[p _(xx)(k)−jq _(xx)(k)]

M _(t) ^((1,2))(k)=βM _(t) ^((1,2))(k−1)+(1−β)[p _(xy)(k)−jq _(xy)(k)]

M _(t) ^((2,1))(k)=βM _(t) ^((2,1))(k−1)+(1−β)[p _(yx)(k)−jq _(yx)(k)]

M _(t) ^((2,2))(k)=βM _(t) ^((2,2))(k−1)+(1−β)[p _(yy)(k)−jq_(yy)(k)]  (39)

where β is a constant and M_(t)(k) can assume an initial value of allzeros. For example, in one embodiment, β=1−2^(−m) (e.g., m=6), althoughother values of β may be used in alternative embodiments.

The cycle slip computation block 804 receives the timing matrixM_(t)(k), and generates a cycle slip number signal ϵ_(cs)(k). The cycleslip number signal ϵ_(cs)(k) represents a number of cycle slips of thesampling phase relative to the received signal. A cycle slip occurs whenthe sampling phase shifts a symbol period T. This effect generallyoccurs during the start-up period and its frequency depends on clockerror between the transmit and receive signals.

To determine the number of cycle slips, the cycle slip computation block804 calculates a determinant of the timing matrix M_(t)(k) as follows:

ρ(k)=M _(t) ^((1,1))(k)*M _(t) ^((2,2))(k)−M _(t) ^((1,2))(k)*M _(t)^((2,1))(k)  (40)

In-phase and quadrature phase error signals and then computed as:

{tilde over (p)}(k)=sign[real{ρ(k)}]

{tilde over (q)}(k)=sign[imag{ρ(k)}]  (41)

where

$\begin{matrix}{{{sign}\lbrack x\rbrack} = \left\{ \begin{matrix}1 & {x \geq 0} \\{- 1} & {x < 0}\end{matrix} \right.} & (42)\end{matrix}$

real[x] is the real part of complex x and imag[x] is the imaginary partof complex x.The accumulated number of cycle slips is determined as:

ϵ_(cs)(k)=Σ_(i=0) ^(k) cs(i)  (43)

where:

cs(k)=−1 when ({tilde over (p)}(k−1),{tilde over (q)}(k−1))=(1,1) and({tilde over (p)}(k),{tilde over (q)}(k))=(1,−1)

cs(k)=−1 when ({tilde over (p)}(k−1),{tilde over (q)}(k−1))=(−1,−1) and({tilde over (p)}(k),{tilde over (q)}(k))=(−1,1)

cs(k)=+1 when ({tilde over (p)}(k−1),{tilde over (q)}(k −1))=(1,−1) and({tilde over (p)}(k),{tilde over (q)}(k))=(1,1)

cs(k)=+1 when ({tilde over (p)}(k−1),{tilde over (q)}(k−1))=(−1,1) and({tilde over (p)}(k),{tilde over (q)}(k))=(−1,−1)

cs(k)=0 otherwise.  (44)

The DTM block 806 computes a phase error signal ϵ_(phase)(k)representing the estimated phase error of a local clock relative to thereceived signal. In the embodiment of FIG. 8A, the phase error signalϵ_(phase)(k) is calculated using a determinant method.

In the determinant method, the phase error signal is given by

ϵ_(phase)(k)=ϵ_(DTM)(k)=γ(k)=arg{ρ(k)}  (45)

In one embodiment, the argument of the determinant is determined using alookup table. In one embodiment, the DTM block 806 and cycle slipcomputation block 804 are integrated such that the determinant of thetiming matrix M_(t)(k) is determined only once, and may then be used tocompute both ϵ_(cs)(k) and ϵ_(phase)(k).

The loop filter 808 filters and combines the phase error signalϵ_(phase)(k) and the cycle slip number signal ϵ_(cs)(k) to generate acontrol signal to the NCO 510 that controls a total accumulated phaseshift in the received signal to be compensated. For example, in oneembodiment, the loop filter 808 applies a proportional-integral (PI)filter to the phase error signal ϵ_(phase)(k) to generate a filteredphase error signal ϵ′_(phase)(k). For example, in one embodiment, the PIfilter has an output L(z)=K_(p)+K_(i)/1−z⁻¹ where K_(p) and K_(i) aregain constants. The loop filter 808 then combines the number of cycleslips ϵ_(cs)(k) with the filtered phase error ϵ′_(phase)(k). to detectthe total rotation and provides this information as a control signal tothe NCO 510. For example, in one embodiment, the loop filter 808implements the function:

NCO_(control)(k)=ϵ′_(phase)(k)−k _(cs)ϵ_(cs)(k)  (46)

where k_(cs) is a constant that represents a gain of the loop filter808. For example, in one embodiment, k_(cs)=2⁻³ although other gainconstants may be used.

FIG. 8B illustrates an alternative embodiment of the rotationcomputation block 508-B which instead uses a modified wave differencemethod to determine the phase error signal ϵ_(phase)(k). In theembodiment of FIG. 8B, the rotation computation block 508-B comprises amatrix estimator 852, a cycle slip computation block 854, a WDM block856, and a loop filter 858. The matrix estimator 852 operates similarlyto the matrix estimator 802 described above but generates both a timingmatrix M_(t)(k) and a noisy timing matrix {circumflex over (M)}_(t)(k).The timing matrix M_(t)(k) is computed in the same manner describedabove. The noisy timing matrix {circumflex over (M)}_(t)(k) is definedas the timing matrix M_(t)(k) where β=0. Thus, the components of{circumflex over (M)}_(t)(k) are given as:

{circumflex over (M)} _(t) ^((1,1))(k)=p _(xx)(k)−jq _(xx)(k)

{circumflex over (M)} _(t) ^((1,2))(k)=p _(xy)(k)−jq _(xy)(k)

{circumflex over (M)} _(t) ^((2,1))(k)=p _(yx)(k)−jq _(yx)(k)

{circumflex over (M)} _(t) ^((2,2))(k)=p _(yy)(k)−jq _(yy)(k)  (47)

The cycle slip computation block 854 generates a cycle slip numbersignal ϵ_(cs)(k) representing the detected number of cycle slips basedon the timing matrix M_(t)(k) in the same manner described above.

The WDM block 856 determines the phase error signal based on thein-phase and quadrature error signals from the in-phase and quadratureerror computation block 506, the timing matrix M_(t)(k), and the noisytiming matrix {circumflex over (M)}_(t)(k) using a modified wavedifference method. In this technique, the phase error signal is computedas:

$\begin{matrix}{\epsilon_{phase} = {{\epsilon_{MWDM}(k)} = {{- }\begin{Bmatrix}{{e^{\frac{j\; {\gamma {(n)}}}{2}}\begin{bmatrix}{{\hat{M}}_{t}^{({1,1})}(k)} & {{\hat{M}}_{t}^{({1,2})}(k)} & {{\hat{M}}_{t}^{({2,1})}(k)} & {{\hat{M}}_{t}^{({2,2})}(k)}\end{bmatrix}} \times} \\\begin{bmatrix}{M_{t}^{({1,1})}(k)} & {M_{t}^{({1,2})}(k)} & {M_{t}^{({2,1})}(k)} & {M_{t}^{({2,2})}(k)}\end{bmatrix}^{H}\end{Bmatrix}}}} & (48)\end{matrix}$

where ℑ{.} denotes the imaginary part, [.]^(H) denotes the complexconjugate and transpose, and

γ(k)=unwrap{γ(k−1),arg(det{M _(t)(k)})}  (49)

where γ(k)∈[−2π, 2π) is the new unwrapped angle based on the oldunwrapped angle and the new argument of the determinant. In more detail,z=unwrap{y, x} computes the unwrapped angle z∈[−2π, 2π) based on theinputs y∈[−2π, 2π), x∈[−π, π) by first adding a multiple of 2π to x togive an intermediate result in the range [y−π, y+π), then adding amultiple of 4π to give a final result z in the range [−2π, 2π). Notethat γ(k) is defined for the range [−2π, 2π) because it is halved whencomputing the phase error. The initial value of γ(k) can be set to 0.

The loop filter 858 operates similarly to the loop filter 808 describedabove.

Although the detailed description contains many specifics, these shouldnot be construed as limiting the scope but merely as illustratingdifferent examples and aspects of the described embodiments. It shouldbe appreciated that the scope of the described embodiments includesother embodiments not discussed in detail above. For example, thefunctionality of the various components and the processes describedabove can be performed by hardware, firmware, software, and/orcombinations thereof.

Various other modifications, changes and variations which will beapparent to those skilled in the art may be made in the arrangement,operation and details of the method and apparatus of the describedembodiments disclosed herein without departing from the spirit and scopeof the invention as defined in the appended claims. Therefore, the scopeof the invention should be determined by the appended claims and theirlegal equivalents.

1. A method for timing recovery comprising: receiving an input signalsampled based on a sampling clock; applying a resonator filter to theinput signal to generate a band pass filtered signal; computing anin-phase and quadrature error signal based on the band pass filteredsignal, the in-phase and quadrature error signal representing an amountof phase error in each of an in-phase component and a quadraturecomponent of the input signal; computing a rotation control signal basedon the in-phase and quadrature error signal, the rotation control signalrepresenting an amount of accumulated phase shift between the inputsignal and the sampling clock; and controlling an oscillator to generatethe sampling clock and to adjust at least one of the phase and frequencyof the sampling clock based on the rotation control signal.
 2. Themethod of claim 1, wherein computing the in-phase and quadrature errorsignal comprises: applying an interpolator filter to the band passfiltered signal to generate an interpolated signal, the interpolatedsignal having a higher sampling rate than the band pass filtered signal;determining a correlation signal representing correlation measuresbetween a first polarization of the interpolated signal and a secondpolarization of the interpolated signal; and determining the in-phaseerror and quadrature error signal based on a difference between thecorrelation measures for symbols of an in-phase component of thecorrelation signal separated by a half symbol period and based on adifference between the correlation measures for symbols of a quadraturecomponent of the correlation signal separated by a half symbol period.3. The method of claim 1, wherein computing the rotation control signalbased on the in-phase and quadrature error signal comprises: determininga timing matrix based on the in-phase and quadrature error signal;determining based on the timing matrix, a cycle slip number signalrepresenting a number of full symbol period shifts of the input signalrelative to the sampling clock; determining based on the timing matrix,a phase error signal representing a difference between the phase theinput signal and the phase of the sampling clock; applying a loop filterto the phase error signal to generate a filtered phase error signal; andgenerating the rotation control signal based on the cycle slip numbersignal and the filtered phase error signal, the rotation control signalto correct for the phase error and the number of full symbol periodshifts of the input signal relative to the sampling clock.
 4. The methodof claim 3, wherein determining the cycle slip number signal comprises:computing a determinant of the timing matrix; and generating the cycleslip number signal based on the determinant.
 5. The method of claim 3,wherein determining the phase error signal based on the timing matrixcomprises: computing a determinant of the timing matrix; and determiningthe phase error based on the determinant.
 6. The method of claim 3,wherein determining the phase error signal based on the timing matrixcomprises: applying a modified wave difference method to the in-phaseand quadrature error signals and the timing matrix to generate the phaseerror signal.
 7. The method of claim 3, wherein the timing matrixrepresents coefficients of a timing tone detected in the input signal,the timing tone representing frequency and phase of a symbol clock ofthe input signal and the timing tone having a non-zero timing toneenergy.
 8. The method of claim 1, wherein the applying the resonatorfilter comprises band pass filtering the input signal around a frequencyof approximately half of the symbol rate of the input signal for eachpolarization of the input signal.
 9. The method of claim 1, wherein theinput signal comprises an impairment introduced in an optical channelincluding at least one: a half symbol period differential group delay, acascaded differential group delay, a dynamic polarization modedispersion, and a residual chromatic dispersion.
 10. A receiver,comprising: an analog front end for receiving an analog input signal andgenerating a digital input signal sampled based on a sampling clock; adigital signal processor to apply a resonator filter to the input signalto generate a band pass filtered signal, to compute an in-phase andquadrature error signal based on the band pass filtered signal, thein-phase and quadrature error signal representing an amount of phaseerror in each of an in-phase component and a quadrature component of theinput signal, and to compute a rotation control signal based on thein-phase and quadrature error signal, the rotation control signalrepresenting an amount of accumulated phase shift between the inputsignal and the sampling clock; and an oscillator controller to controlan oscillator to adjust at least one of the phase and frequency of thesampling clock based on the rotation control signal.
 11. The receiver ofclaim 10, wherein the in-phase and quadrature error computation blockcomprises: an interpolator filter to filter the band pass filteredsignal to generate an interpolated signal, the interpolated signalhaving a higher sampling rate than the band pass filtered signal; acorrelation computation block to generate a correlation signal based onthe interpolated signal, the correlation signal representing correlationmeasures between a first polarization of the interpolated signal and asecond polarization of the interpolated signal; and a differencecomputation block to generate the in-phase and quadrature error signalbased on a difference between the correlation measures for symbols of anin-phase component of the correlation signal separated by a half symbolperiod, and based on a difference between the correlation measures forsymbols of a quadrature component of the correlation signal separated bya half symbol period.
 12. A receiver of claim 10, wherein the phaseerror computation block comprises: a timing matrix estimation block togenerate a timing matrix based on the in-phase and quadrature errorsignal; a cycle slip computation block to determine based on the timingmatrix, a cycle slip number signal representing a number of full symbolperiod shifts of the input signal relative to the sampling clock; aphase error computation block to generate based on the timing matrix, aphase error signal representing a phase error of the input signalrelative to the sampling clock; and a loop filter to filter the phaseerror signal to generate a filtered phase error signal and to generatethe rotation control signal based on the cycle slip number signal andthe filtered phase error signal, the rotation control signal to correctfor the phase error and the number of full symbol period shifts of theinput signal relative to the sampling clock.
 13. The receiver of claim12, wherein the cycle slip computation block is further configured to:compute a determinant of the timing matrix; and generate the cycle slipnumber signal based on the determinant.
 14. The receiver of claim 12,wherein the phase error computation block is further configured to:compute a determinant of the timing matrix; and determine the phaseerror based on the determinant.
 15. The receiver of claim 12, whereinthe phase error computation block is further configured to: apply amodified wave difference method to the in-phase and quadrature errorsignals and the timing matrix to generate the phase error signal. 16.The receiver of claim 12, wherein the timing matrix representscoefficients of a timing tone detected in the input signal, the timingtone representing frequency and phase of a symbol clock of the inputsignal and the timing tone having a non-zero timing tone energy.
 17. Areceiver, comprising: an optical front end to receive an optical signalfrom an optical communication channel and to convert the optical signalto an analog electrical signal; an analog front end the receive theanalog electrical signal and to sample the analog electrical signalbased on a sampling clock to generate a digital signal; an oscillator togenerate the sampling clock; a digital signal processor to apply aresonator filter to the input signal to generate a band pass filteredsignal, to compute an in-phase and quadrature error signal based on theband pass filtered signal, the in-phase and quadrature error signalrepresenting an amount of phase error in each of an in-phase componentand a quadrature component of the input signal, and to compute arotation control signal based on the in-phase and quadrature errorsignal, the rotation control signal representing an amount ofaccumulated phase shift between the input signal and the sampling clock;an oscillator controller to control the oscillator to adjust at leastone of the phase and frequency of the sampling clock based on therotation control signal; a demodulator to demodulate the digital signalto generate a demodulated signal; and a decoder to decode thedemodulated signal to generate a decoded output signal.
 18. The receiverof claim 17, wherein the in-phase and quadrature error computation blockcomprises: an interpolator filter to filter the band pass filteredsignal to generate an interpolated signal, the interpolated signalhaving a higher sampling rate than the band pass filtered signal; acorrelation computation block to generate a correlation signal based onthe interpolated signal, the correlation signal representing correlationmeasures between a first polarization of the interpolated signal and asecond polarization of the interpolated signal; and a differencecomputation block to generate the in-phase and quadrature error signalbased on a difference between the correlation measures for symbols of anin-phase component of the correlation signal separated by a half symbolperiod, and based on a difference between the correlation measures forsymbols of a quadrature component of the correlation signal separated bya half symbol period.
 19. A receiver of claim 17, wherein the phaseerror computation block comprises: a timing matrix estimation block togenerate a timing matrix based on the in-phase and quadrature errorsignal; a cycle slip computation block to determine based on the timingmatrix, a cycle slip number signal representing a number of full symbolperiod shifts of the input signal relative to the sampling clock; aphase error computation block to generate based on the timing matrix, aphase error signal representing a phase error of the input signalrelative to the sampling clock; and a loop filter to filter the phaseerror signal to generate a filtered phase error signal and to generatethe rotation control signal based on the cycle slip number signal andthe filtered phase error signal, the rotation control signal to correctfor the phase error and the number of full symbol period shifts of theinput signal relative to the sampling clock.
 20. The receiver of claim17, wherein the timing matrix represents coefficients of a timing tonedetected in the input signal, the timing tone representing frequency andphase of a symbol clock of the input signal and the timing tone having anon-zero timing tone energy.